OUTLINES

1. Modal methods for 3D heterogeneous neutronics core calculations using the mixed dual solver MINOS. Application to complex geometries and parallel processing. Pierre Guérin, Jean-Jacques Lautard pierre.guerin@cea.fr CEA SACLAY DEN/DANS/DM2S/SERMA/LENR 91191 Gif sur Yvette Cedex

2. OUTLINES • General considerations and motivations • The component mode synthesis method • A factorized component mode synthesis method • Parallelization • Conclusions and perspectives

3. General considerations and motivations The component mode synthesis method A factorized component mode synthesis method Parallelization Conclusions and perspectives

4. Pin assembly Core Pin by pin geometry Cell by cell mesh Whole core mesh Geometry and mesh of a PWR 900 MWe core

5. Introduction and motivations • MINOS solver : • main core solver of the DESCARTES system, developed by CEA, EDF and AREVA • mixed dual finite element method for the resolution of the equations in 3D cartesian homogenized geometries • 3D cell by cell homogenized calculations currently expensive • Standard reconstruction techniques to obtain the local pin power can be improved for MOX reloaded cores • interface between UOX and MOX assemblies • Motivations: • Find a numerical method that takes in account the heterogeneity of the core • Perform calculations on parallel computers

6. General considerations and motivations The component mode synthesis method A factorized component mode synthesis method Parallelization Conclusions and perspectives

7. The CMS method • CMS method for the computation of the eigenmodes of partial differential equations has been used for a long time in structural analysis. • The steps of the CMS method : • Decomposition of the domain in K subdomains • Calculation of the first eigenfunctions of the local problem on each subdomain • All these local eigenfunctions span a discrete space used for the global solve by a Galerkin technique

8. : Current : Flux Monocinetic diffusion model • Monocinetic diffusion eigenvalue problem with homogeneous Dirichlet boundary condition: Fundamental eigenvalue • Mixed dual weak formulation : find such that

9. Local eigenmodes • Overlapping domain decomposition : • Computation on each of the first local eigenmodes with the global boundary condition on , and on \ :

10. Global Galerkin method • Extension on R by 0 of the local eigenmodes on each :  global functional spaces on R • Global eigenvalue problem on these spaces :

11. Linear system and • Unknowns : • Linear system associated : with : If all the integrals over vanish  sparse matrices

12. Global problem • Global problem : • H symmetric but not positive definite

13. Domain decomposition • Domain decomposition in 201 subdomains for a PWR 900 MWe loaded with UOX and MOX assemblies : • Internal subdomains boundaries : • on the middle of the assemblies • condition is close to the real value • Interface problem between UOX and MOX is avoided

14. Power and scalar flux representation • diffusion calculation • two energy groups • cell by cell mesh • RTo element Thermal flux Fast flux Power in the core

15. Comparison between CMS method and MINOS • keff difference, and norm of the power difference between CMS method and MINOS solution Two CMS method cases : • 4 flux and 6 curent modes on each subdomain • 9 flux and 11 current modes on each subdomain • More current modes than flux modes

16. Comparison between CMS method and MINOS • Power gap between CMS method and MINOS in the two cases. 1% 5% 0% 0% -1% -5% 9 flux modes, 11 current modes 4 flux modes, 6 current modes

17. Comparison between CMS method and MINOS • Power cell difference between CMS method and MINOS solution in the two cases. Total number of cells : 334084. 9 flux modes, 11 current modes 95% of the cells : power gap < 0,1% 4 flux modes, 6 current modes 95% of the cells : power gap < 1%

18. General considerations and motivations The component mode synthesis method A factorized component mode synthesis method Parallelization Conclusions and perspectives

19. Factorization principle • Goal: decrease CPU time and memory storage  only the fundamental mode calculation  replace the higher order modes by suitably chosen functions • Factorization principle on a periodic core : • is a smooth function solution of a homogenized diffusion problem: • is the local fundamental solution on an assembly of the problem with infinite medium boundary conditions • We adapt this principle on a non periodic core in order to replace the higher order modes

20. The factorized CMS method : FCMS • solution of the problem:  analytical solution  sines or cosines • the fundamental eigenmode on each subdomain. • New current basis functions: • New flux basis functions:

21. Comparison between FCMS method and MINOS 0 • Same domain decomposition • 6 flux modes and 11 current modes • Differences between FCMS and MINOS in 2D : 97% of the cells  power gap < 1%

22. JHR research reactor: first result • 9 subdomains • : not yet a satisfactory result  improve the domain decomposition

23. JHR: power distribution

24. JHR: flux for the 6 energy groups • Thermal flux • Fast flux

25. General considerations and motivations The component mode synthesis method A factorized component mode synthesis method Parallelization Conclusions and perspectives

26. Parallelization of our methods in 3D • Most of the calculation time: local solves and matrix calculation • Local solves are independent, no communication • Matrix calculations are parallelized with communications between the close subdomains • Global resolution: very fast, sequential

27. General considerations and motivations The component mode synthesis method A factorized component mode synthesis method Parallelization Conclusions and perspectives

28. Conclusions and perspectives • Modal synthesis method : • Good accuracy for the keff and the local cell power • Well fitted for parallel calculation: • the local calculations are independent • they need no communication • Future developments : • Extension to 3D cell by cell calculations • Another geometries (EPR, HTR…) • Pin by pin calculation • Time dependent calculations • Coupling local calculation and global diffusion resolution • Complete transport calculations